function F = sd_log_rev_int(k_j, sigma)

fun1= @(x, k) log((x.^(-1))-1).*((x .*exp(x-1)).^(k-1)).* ((1+x).*exp(x-1));
Integral1 = integral(@(x)fun1(x,k_j),0,1);

fun2= @(x, k) ((log(((x .*exp(x-1)).*((x.^(-1))-1)))- (-(1./k)+(k.*Integral1))).^(2))   .* k   .*((x .*exp(x-1)).^(k-1))  .* ((1+x).*exp(x-1));    
Integral2 = integral(@(x)fun2(x,k_j),0,1);

sigmatheory = sqrt(Integral2);
F = sigmatheory-sigma;